import collections

Set = set

try:
    from collections import OrderedDict
except ImportError:
    class OrderedDict(dict):
        'Dictionary that remembers insertion order'
        # An inherited dict maps keys to values.
        # The inherited dict provides __getitem__, __len__, __contains__, and get.
        # The remaining methods are order-aware.
        # Big-O running times for all methods are the same as for regular dictionaries.

        # The internal self.__map dictionary maps keys to links in a doubly linked list.
        # The circular doubly linked list starts and ends with a sentinel element.
        # The sentinel element never gets deleted (this simplifies the algorithm).
        # Each link is stored as a list of length three:  [PREV, NEXT, KEY].

        def __init__(self, *args, **kwds):
            '''Initialize an ordered dictionary.  Signature is the same as for
            regular dictionaries, but keyword arguments are not recommended
            because their insertion order is arbitrary.

            '''
            if len(args) > 1:
                raise TypeError('expected at most 1 arguments, got %d' % len(args))
            try:
                self.__root
            except AttributeError:
                self.__root = root = []                     # sentinel node
                root[:] = [root, root, None]
                self.__map = {}
            self.__update(*args, **kwds)

        def __setitem__(self, key, value, dict_setitem=dict.__setitem__):
            'od.__setitem__(i, y) <==> od[i]=y'
            # Setting a new item creates a new link which goes at the end of the linked
            # list, and the inherited dictionary is updated with the new key/value pair.
            if key not in self:
                root = self.__root
                last = root[0]
                last[1] = root[0] = self.__map[key] = [last, root, key]
            dict_setitem(self, key, value)

        def __delitem__(self, key, dict_delitem=dict.__delitem__):
            'od.__delitem__(y) <==> del od[y]'
            # Deleting an existing item uses self.__map to find the link which is
            # then removed by updating the links in the predecessor and successor nodes.
            dict_delitem(self, key)
            link_prev, link_next, key = self.__map.pop(key)
            link_prev[1] = link_next
            link_next[0] = link_prev

        def __iter__(self):
            'od.__iter__() <==> iter(od)'
            root = self.__root
            curr = root[1]
            while curr is not root:
                yield curr[2]
                curr = curr[1]

        def __reversed__(self):
            'od.__reversed__() <==> reversed(od)'
            root = self.__root
            curr = root[0]
            while curr is not root:
                yield curr[2]
                curr = curr[0]

        def clear(self):
            'od.clear() -> None.  Remove all items from od.'
            try:
                for node in self.__map.itervalues():
                    del node[:]
                root = self.__root
                root[:] = [root, root, None]
                self.__map.clear()
            except AttributeError:
                pass
            dict.clear(self)

        def popitem(self, last=True):
            '''od.popitem() -> (k, v), return and remove a (key, value) pair.
            Pairs are returned in LIFO order if last is true or FIFO order if false.

            '''
            if not self:
                raise KeyError('dictionary is empty')
            root = self.__root
            if last:
                link = root[0]
                link_prev = link[0]
                link_prev[1] = root
                root[0] = link_prev
            else:
                link = root[1]
                link_next = link[1]
                root[1] = link_next
                link_next[0] = root
            key = link[2]
            del self.__map[key]
            value = dict.pop(self, key)
            return key, value

        # -- the following methods do not depend on the internal structure --

        def keys(self):
            'od.keys() -> list of keys in od'
            return list(self)

        def values(self):
            'od.values() -> list of values in od'
            return [self[key] for key in self]

        def items(self):
            'od.items() -> list of (key, value) pairs in od'
            return [(key, self[key]) for key in self]

        def iterkeys(self):
            'od.iterkeys() -> an iterator over the keys in od'
            return iter(self)

        def itervalues(self):
            'od.itervalues -> an iterator over the values in od'
            for k in self:
                yield self[k]

        def iteritems(self):
            'od.iteritems -> an iterator over the (key, value) items in od'
            for k in self:
                yield (k, self[k])

        def update(*args, **kwds):
            '''od.update(E, **F) -> None.  Update od from dict/iterable E and F.

            If E is a dict instance, does:           for k in E: od[k] = E[k]
            If E has a .keys() method, does:         for k in E.keys(): od[k] = E[k]
            Or if E is an iterable of items, does:   for k, v in E: od[k] = v
            In either case, this is followed by:     for k, v in F.items(): od[k] = v

            '''
            if len(args) > 2:
                raise TypeError('update() takes at most 2 positional '
                                'arguments (%d given)' % (len(args),))
            elif not args:
                raise TypeError('update() takes at least 1 argument (0 given)')
            self = args[0]
            # Make progressively weaker assumptions about "other"
            other = ()
            if len(args) == 2:
                other = args[1]
            if isinstance(other, dict):
                for key in other:
                    self[key] = other[key]
            elif hasattr(other, 'keys'):
                for key in other.keys():
                    self[key] = other[key]
            else:
                for key, value in other:
                    self[key] = value
            for key, value in kwds.items():
                self[key] = value

        __update = update  # let subclasses override update without breaking __init__

        __marker = object()

        def pop(self, key, default=__marker):
            '''od.pop(k[,d]) -> v, remove specified key and return the corresponding value.
            If key is not found, d is returned if given, otherwise KeyError is raised.

            '''
            if key in self:
                result = self[key]
                del self[key]
                return result
            if default is self.__marker:
                raise KeyError(key)
            return default

        def setdefault(self, key, default=None):
            'od.setdefault(k[,d]) -> od.get(k,d), also set od[k]=d if k not in od'
            if key in self:
                return self[key]
            self[key] = default
            return default

        def __repr__(self, _repr_running={}):
            'od.__repr__() <==> repr(od)'
            call_key = id(self), _get_ident()
            if call_key in _repr_running:
                return '...'
            _repr_running[call_key] = 1
            try:
                if not self:
                    return '%s()' % (self.__class__.__name__,)
                return '%s(%r)' % (self.__class__.__name__, self.items())
            finally:
                del _repr_running[call_key]

        def __reduce__(self):
            'Return state information for pickling'
            items = [[k, self[k]] for k in self]
            inst_dict = vars(self).copy()
            for k in vars(OrderedDict()):
                inst_dict.pop(k, None)
            if inst_dict:
                return (self.__class__, (items,), inst_dict)
            return self.__class__, (items,)

        def copy(self):
            'od.copy() -> a shallow copy of od'
            return self.__class__(self)

        @classmethod
        def fromkeys(cls, iterable, value=None):
            '''OD.fromkeys(S[, v]) -> New ordered dictionary with keys from S
            and values equal to v (which defaults to None).

            '''
            d = cls()
            for key in iterable:
                d[key] = value
            return d

        def __eq__(self, other):
            '''od.__eq__(y) <==> od==y.  Comparison to another OD is order-sensitive
            while comparison to a regular mapping is order-insensitive.

            '''
            if isinstance(other, OrderedDict):
                return len(self)==len(other) and self.items() == other.items()
            return dict.__eq__(self, other)

        def __ne__(self, other):
            return not self == other

        # -- the following methods are only used in Python 2.7 --

        def viewkeys(self):
            "od.viewkeys() -> a set-like object providing a view on od's keys"
            return KeysView(self)

        def viewvalues(self):
            "od.viewvalues() -> an object providing a view on od's values"
            return ValuesView(self)

        def viewitems(self):
            "od.viewitems() -> a set-like object providing a view on od's items"
            return ItemsView(self)

KEY, PREV, NEXT = range(3)

class OrderedSet(collections.MutableSet):
  """
  From: http://code.activestate.com/recipes/576694/
  """
  def __init__(self, iterable=None):
    self.end = end = []
    end += [None, end, end]         # sentinel node for doubly linked list
    self.map = {}                   # key --> [key, prev, next]
    if iterable is not None:
      self |= iterable

  def __len__(self):
    return len(self.map)

  def __contains__(self, key):
    return key in self.map

  def add(self, key):
    if key not in self.map:
      end = self.end
      curr = end[PREV]
      curr[NEXT] = end[PREV] = self.map[key] = [key, curr, end]

  def discard(self, key):
    if key in self.map:
      key, prev, next = self.map.pop(key)
      prev[NEXT] = next
      next[PREV] = prev

  def __iter__(self):
    end = self.end
    curr = end[NEXT]
    while curr is not end:
      yield curr[KEY]
      curr = curr[NEXT]

  def __reversed__(self):
    end = self.end
    curr = end[PREV]
    while curr is not end:
      yield curr[KEY]
      curr = curr[PREV]

  def pop(self, last=True):
    if not self:
      raise KeyError('set is empty')
    key = next(reversed(self)) if last else next(iter(self))
    self.discard(key)
    return key

  def __repr__(self):
    if not self:
      return '%s()' % (self.__class__.__name__,)
    return '%s(%r)' % (self.__class__.__name__, list(self))

  def __eq__(self, other):
    if isinstance(other, OrderedSet):
      return len(self) == len(other) and list(self) == list(other)
    return set(self) == set(other)

  def __del__(self):
    self.clear()                    # remove circular references
